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The Journal of Experimental Biology 214, 263-274 © 2011. Published by The Company of Biologists Ltd doi:10.1242/jeb.047985

Molecules in motion: influences of diffusion on metabolic structure and function in skeletal muscle

Stephen T. Kinsey1,*, Bruce R. Locke2 and Richard M. Dillaman1 1Department of Biology and Marine Biology, University of North Carolina Wilmington, 601 South College Road, Wilmington, NC 28403-5915, USA and 2Department of Chemical and Biomedical Engineering, Florida State University, FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA *Author for correspondence ([email protected])

Accepted 25 August 2010

Summary Metabolic processes are often represented as a group of metabolites that interact through enzymatic reactions, thus forming a network of linked biochemical pathways. Implicit in this view is that diffusion of metabolites to and from enzymes is very fast compared with reaction rates, and metabolic fluxes are therefore almost exclusively dictated by catalytic properties. However, diffusion may exert greater control over the rates of reactions through: (1) an increase in reaction rates; (2) an increase in diffusion distances; or (3) a decrease in the relevant diffusion coefficients. It is therefore not surprising that skeletal muscle fibers have long been the focus of reaction–diffusion analyses because they have high and variable rates of ATP turnover, long diffusion distances, and hindered metabolite diffusion due to an abundance of intracellular barriers. Examination of the diversity of skeletal muscle fiber designs found in animals provides insights into the role that diffusion plays in governing both rates of metabolic fluxes and cellular organization. Experimental measurements of metabolic fluxes, diffusion distances and diffusion coefficients, coupled with reaction–diffusion mathematical models in a range of muscle types has started to reveal some general principles guiding muscle structure and metabolic function. Foremost among these is that metabolic processes in muscles do, in fact, appear to be largely reaction controlled and are not greatly limited by diffusion. However, the influence of diffusion is apparent in patterns of fiber growth and metabolic organization that appear to result from selective pressure to maintain reaction control of metabolism in muscle. Key words: calcium, diffusion, metabolism, mitochondria, muscle, nuclei.

Introduction in D values, diffusion distances and rates of reaction fluxes. For Physiologists have a long history of evaluating the role of diffusion instance, Ca2+ cycling entails a high D for Ca2+, short diffusion in cellular metabolism, particularly with respect to the necessity for distances and high reaction fluxes, whereas nuclear function entails O2 transport from the blood into cells for aerobic metabolism. low D values for various macromolecular products, long diffusion Indeed, Fick’s diffusion equations and principles of membrane distances and low reaction fluxes. These interactions mean that cell transport have a prominent position in most physiology textbooks, structure is responsive to diffusion constraints in a manner that is and biology students are well versed in the notion that cells are dependent on the type of process and the functional demands placed small in order to maintain short diffusion distances for molecules on that process. Fig.1 also illustrates the principle of facilitated like O2. A corollary of this rule in skeletal muscle is that aerobic diffusion that was first proposed for myoglobin (Mb) by Wittenberg fibers that rely on O2 diffusion to promote sustained exercise are (Wittenberg, 1959) and independently identified for hemoglobin generally smaller than anaerobic fibers that are used for burst (Hb) by Scholander (Scholander, 1960), but also applies to contraction and are less reliant on O2 diffusion (for review, see van phosphagen kinases such as creatine kinase (CK) and arginine Wessel et al., 2010). The implication of this observation is that kinase (AK), and to a lesser extent to parvalbumin (PA). In higher rates of aerobic metabolism require smaller fibers, and facilitated diffusion, protein binding or enzymatic conversion of the variation in fiber size may represent responses to avoid diffusion diffusing species provides a parallel pathway for diffusive flux that limitation. In a more general sense, we can conclude that in any enhances the overall rate of diffusive transport. The present paper reaction–diffusion system, the role of diffusion becomes greater as will attempt to summarize some of the ways in which diffusion the diffusion coefficient (D) decreases, diffusion distance increases governs metabolic structure and function in skeletal muscle. or the rate of reaction flux increases (Weisz, 1973). In principle, nearly every biochemical reaction is a Concentration gradients are a prerequisite for diffusion reaction–diffusion process involving the diffusive flux of substrates control of reaction flux to enzyme active sites. A number of processes have been the When examining a single tissue type, D is comparatively invariant subject of reaction–diffusion analyses in muscle, such as the over physiological time scales, so the extent to which diffusion aforementioned oxygen flux from capillaries to mitochondria influences cell structure and function is largely governed by the (Fig.1A), the diffusion of ATP to sites of cellular ATPases interaction between diffusion distance and reaction flux rate. In (Fig.1B), Ca2+ cycling during contraction–relaxation cycles skeletal muscle fibers rates of reaction fluxes and diffusion (Fig.1C), and the transport of nuclear products to sites of action in distances can vary over several orders of magnitude. For instance, the cell (Fig.1D). These examples illustrate considerable variation aerobic metabolic rate in white muscle from fishes and crustaceans

THE JOURNAL OF EXPERIMENTAL BIOLOGY 264 S. T. Kinsey, B. R. Locke and R. M. Dillaman

these approaches is that diffusion is rapid relative to the rate of metabolic flux, and it therefore does not limit reactions. A Fig.2 is a diagram of the manner in which diffusion distance and MbO MbO reaction flux interact to affect concentration profiles of a diffusing HbO2 2 2 Net diffusive molecule for cases where diffusion would not be limiting and for flux O O O cases where it may be limiting. In this example, there is a point +2 +2 +2 Hb Mb Mb source for the diffusing molecule and that molecule is consumed (sink) as it diffuses away from the source. For instance, the source 2H2O Capillary Muscle fiber could be a capillary supplying O2 that diffuses across the cell and B is consumed by mitochondria. The schematic diagram in Fig.2 ADP ADP illustrates that in cases where diffusion is fast relative to the +Pi AP/PCr AP/PCr +P reaction rate, there are no concentration gradients for the diffusing ½ O2 i AK/CK Net diffusive flux AK/CK ATPase species (red lines). This represents a situation where diffusion would have no effect on reaction rate, and an analysis of the H2O Arg/Cr Arg/Cr ATP ATP catalytic properties alone is sufficient to explain the metabolic Mitochondria process. Thus, an increase in the reaction rate or the diffusion distance (manifested here as increased activity or size, respectively, of the sink) would lead to a uniform reduction in the concentration C CsCa2+ PACa2+ PACa2+ TnCCa2+ ATP of the diffusing species over the diffusion distance. The blue lines Net diffusive flux illustrate what would be expected when diffusion is not much faster Ca2+ Ca2+ Ca2+ Ca2+ + + + + than the rate of reaction flux, leading to concentration gradients that Cs PA PA TnC become steeper as the reaction rate or the diffusion distance Sarcoplasmic increases. In these cases, if the diffusing species is a substrate for reticulum a reaction, then the rate of product formation may be reduced as the distance from the source increases. D Unfortunately, measurements of metabolite concentrations in cells typically cannot distinguish between the rapid diffusion and RNA, Net diffusive flux or Sites of proteins active transport action slow diffusion cases, as they do not give information on spatial variation in concentration. Most studies have therefore analyzed reaction–diffusion processes in muscle using mathematical models Nucleus that include independent measurements of D, diffusion distances and rates of reaction fluxes. This requires a clear understanding of Fig.1. Examples of reaction–diffusion processes in muscle. (A)O2 diffusion the nature of the intracellular environment and the manner in which entails reversible binding with hemoglobin (Hb) or another blood pigment small and large molecules move in this environment. type, and with myoglobin (Mb) in some fiber types. (B)Diffusion of ATP to cellular ATPases involves the reversible transfer of a phosphoryl group from ATP to an acceptor molecule such as creatine (Cr) or arginine (Arg), The intracellular environment of muscle has characteristics of forming the phosphagen phosphocreatine (PCr) or arginine phosphate a porous medium (AP), respectively. Phosphagen kinases, such as creatine kinase (CK) or The cytoplasm is a complex and crowded medium consisting of arginine kinase (AK) catalyze these reactions. (C)Ca2+ is released from the soluble and bound macromolecules, fibrous cytoskeletal elements sarcoplasmic reticulum (SR) upon muscle stimulation and must diffuse to and membrane-bound organelles (reviewed in Luby-Phelps, 2000; and bind myofibrillar troponin C (TnC) to activate contraction. Relaxation Saks et al., 2008). Skeletal muscle has a highly regular and ordered entails the uptake of Ca2+ into the SR, where reversible binding to calsequestrin (Cs) may occur. In some fiber types, parvalbumin (PA) intracellular environment where most of the volume is devoted to reversibly binds Ca2+ that is in the sarcoplasm. In addition, ATP binds Ca2+ the myofibrils, sarcoplasmic reticulum (SR) and mitochondria. and may further facilitate diffusion (not shown). (D)Transcription and Wheatley and colleagues have argued that classical diffusion theory translation entails diffusive flux of nuclear products to various sites of action should not be applied to intracellular movement because (1) many in the cell. In all cases, the red arrows indicate protein-mediated binding, of its assumptions may be violated in the cytoplasm, (2) much of transport or catalysis, while the black arrows indicate diffusion. The intracellular transport is ‘directed’ rather than random, and (3) the thickness of the arrows indicates the relative importance of the diffusive pathway. Dashed lines indicate that Mb and PA are not present in all fiber cytoplasmic structure is incompletely characterized and temporally types. dynamic, and our understanding of interactions of mobile species within this internal structure is inadequate to permit mathematical evaluation (Agutter et al., 1995; Wheatley, 2003). These concerns can be <0.1mol ATPg–1min–1 (Kinsey et al., 2005; Nyack et al., appear to be greatest for macromolecules that may be present 2007), whereas in insect flight muscle it can exceed 2000mol in relatively low concentrations, actively transported along ATPg–1min–1 (Suarez, 1998). Similarly, diffusion distances within cytoskeletal elements in some cell types, and subject to specific fibers can vary from <1m to several hundred micrometers (Kinsey binding interactions. While we appreciate these points, we would et al., 2007). At issue is whether an intracellular process leads to argue that the experimental evidence, some of which is summarized the formation of concentration gradients. Most efforts to model here, indicates that much of molecular motion in muscle fibers is metabolism in muscle have focused exclusively on kinetic true diffusion that can be evaluated mathematically, as is common properties, and these approaches have successfully predicted practice in engineering literature when describing molecular metabolic fluxes over a range of physiological states (e.g. Vicini movement in porous media such as gels (reviewed in Locke, 2001). and Kushmerick, 2000; Lambeth and Kushmerick, 2002; This is particularly true for large, anaerobic fibers from organisms Korzeniewski, 2003; Beard, 2005). An underlying assumption of such as fishes and crustaceans, where, for example, the few

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Fig.2. Schematic diagram of the interaction between Source Sink diffusion distance and reaction flux rate in concentration profiles for cases where diffusion is very rapid compared Diffusion distance with the flux rate (red lines) and cases where diffusion is not much more rapid than the flux rate (blue lines). The red lines therefore represent cases where diffusion can Constant diffusion distance Constant reaction flux rate be ignored and catalytic properties exert complete control over reaction flux. Cases with higher rates of reaction Rapid diffusion flux or longer diffusion distances are indicated by dashed Slow diffusion lines. The left panel shows the influence of changes in the reaction flux when diffusion distance is constant and Low rate of reaction flux Short diffusion distance reaction flux varies, and the right panel shows the influence of diffusion distance when the reaction flux is constant and diffusion distance varies. See text for a tion additional details.

Concentr High rate of reaction flux Long diffusion distance

Distance from source mitochondria in the fiber produce ATP that must move long a range of metabolites and fiber types. For instance, intracellular D distances through the myofibrillar lattice (discussed below). There values of small metabolites like phosphagens, ATP, lactate and is no evidence for metabolic compartmentalization (as has been alanine have been found to be approximately 2-fold lower than in proposed for mammalian cardiomyocytes) (see Saks et al., 2008) aqueous solution in isolated muscle from frog (Yoshizaki et al., or targeted transport in this type of cell, and diffusion seems the 1982), fish (Hubley et al., 1995; Hubley and Moerland, 1995; most likely mechanism for equilibrating ATP in the fiber. Thus, in Kinsey et al., 1999) and crustaceans (Kinsey and Ellington, 1996; this paper we will use the term ‘diffusion’, which can be Kinsey and Moerland, 2002). Similar results have been seen in characterized by D, where it seems to apply based on the mammalian skeletal muscle in vivo (Moonen et al., 1990; van experimental data. Gelderen et al., 1994; de Graaf et al., 2000; de Graaf et al., 2001). Methods of measurement of intracellular diffusion in muscle are Soluble proteins increase the viscosity of solutions like the largely dependent on the type of molecule being observed. cytoplasm, and the ordering of water on hydrophobic surfaces or Diffusion of gases through tissues has been the subject of study for in hydration shells around charged intracellular components might nearly 100 years, and typically entails measurement of partial also reduce mobility in the cytoplasm. Both of these effects might pressures on either side of a tissue through which diffusion is be expected to lead to the observed reduction in the intracellular D occurring. Diffusion of small metabolites and ions in muscle has of metabolites compared with that in aqueous solution. However, been measured using isotopic labeling (Kushmerick and Podolsky, careful measurements have shown that while the shear viscosity of 1969) and reversibly binding fluorophores (Bernengo et al., 2001), the cytoplasm may be relatively high (making it more resistant to but the majority of studies in muscle have used pulsed-field flow than bulk water), the intracellular solvent viscosity is low and gradient nuclear magnetic resonance (NMR) or diffusion-sensitive similar to that of bulk water (Fushimi and Verkman, 1991; Luby- magnetic resonance imaging (MRI) (reviewed in Nicolay et al., Phelps et al., 1993). This suggests that the reduced D of metabolites 2001). Diffusion of macromolecules such as proteins is difficult to inside muscle fibers is the result of intracellular structures that measure using NMR because the relatively slow rotational motion hinder motion (including soluble proteins) rather than the inherent of these molecules causes severe broadening of spectral peaks, viscous properties of intracellular water. although myoglobin diffusion in muscle has been measured using Diffusion of macromolecules in muscle is also reduced this approach (Livingston et al., 1983; Lin et al., 2007). Therefore, compared with that in aqueous solution, but to a much greater fluorescent-labeling approaches, principally fluorescence recovery extent than in small molecules. The diffusion of Mb in skeletal after photobleaching (FRAP) techniques, have been the mostly muscle has been of particular interest because of its role in widely used means of measuring macromolecular diffusion in reversibly binding O2 and therefore serving as a temporary O2 store muscle (reviewed in Verkman, 2003). and facilitating O2 diffusion (reviewed in Wittenberg and Bunch and Kallsen measured diffusion of water, urea and Wittenberg, 2003). Several studies have found that D of Mb in glycerol in barnacle muscle fibers and found, surprisingly, that muscle fibers is about 1/6 to 1/10 of that in aqueous solution there were no differences between D in the muscle sarcoplasm and (Baylor and Pape, 1988; Jurgens et al., 1994; Papadopoulos et al., in an aqueous solution (Bunch and Kallsen, 1969). Kushmerick and 1995; Papadopoulos et al., 2000; Papadopoulos et al., 2001). Podolsky, in contrast, found that a variety of ionic and non-ionic Measurements of protein diffusion in skeletal muscle from frog species of low molecular mass had D values for diffusion along the (Maughan and Lord, 1988; Maughan and Godt, 1999) and cultured length of frog muscle fibers that were reduced by approximately mammalian fibers (Arrio-Dupont et al., 1997; Arrio-Dupont et al., 50% from those seen in aqueous solution (Kushmerick and 2000) yielded D values that were minimally 1/3 lower than that in Podolsky, 1969). This difference was attributed to intracellular water, and in many cases much more dramatically reduced. structures that hinder diffusion and the relatively high protein Analyses of the influence of hydrodynamic radius on the D values concentration in cells, and similar results have since been found for of macromolecules demonstrated that for both proteins

THE JOURNAL OF EXPERIMENTAL BIOLOGY 266 S. T. Kinsey, B. R. Locke and R. M. Dillaman

100 the square root of molecular mass (not shown), rather than the cube A root as would be expected from the Stokes–Einstein relationship 10 (Fig.3B). Arrio-Dupont and colleagues attributed this discrepancy to the fact that the Stokes–Einstein relationship describes diffusion 1 of a hard sphere, whereas real molecules are not spherical or necessarily compact (Arrio-Dupont et al., 1996), but we might also 0.1 conclude that increased steric effects in macromolecules might further alter the relationship between D and molecular mass in 0.01 muscle when evaluated over a large size range of molecules.

) It should be noted that we have not included in Fig.3 diffusing –1 species that are known to be bound or compartmentalized in the s 0.001 2 10 100 1000 10,000 100,000 1000,000 fiber, such as Ca2+, or those macromolecules that are so large as to cm Molecular mass (Da) be essentially immobile, because these conditions lead to a much –6 12 reduced apparent D (Kushmerick and Podolsky, 1969; Arrio- B 10 Dupont et al., 2000; Papadopolous et al., 2000). This is not an effort D ( ϫ 10 to deemphasize the importance of specific or non-specific binding, 8 enzyme localization or potential compartmentalization. Rather, we 6 are summarizing the movement of intracellular molecules and other probes where D has been measured without these confounding 4 effects, in order to evaluate the nature of the intrafiber environment. 2 In addition, it is possible that intracellular convection associated with contraction may enhance intracellular transport (Hochachka, 0 1999; Suarez et al., 2003). There has been limited experimental –2 testing of this hypothesis, although the D of several proteins was 0 0.1 0.2 0.3 0.4 not increased by muscle contraction or passive stretching and 1 shortening (Baylor and Pape, 1988; Papadopoulos et al., 2000). To 3 Mass our knowledge the influence of contraction on the diffusion of small metabolites has not been addressed, and further study in this Fig.3. Relationship between the diffusion coefficient (D) in muscle and area is warranted. molecular mass for a diversity of skeletal muscle and molecule types. D values have been corrected to a temperature (T) of 20°C using a Q10 for Diffusion in muscle is orientation dependent (for small diffusion in muscle of 1.28 (Hubley et al., 1995). Only freely diffusing molecules) species are plotted here. Molecules that are bound or compartmentalized in One of the hallmarks of muscle intracellular structure is that it is the fiber (e.g. Ca2+) or those macromolecules or complexes (e.g. polysomes) that are too large to move through the fiber have lower D highly orientation dependent (anisotropic), most notably with values than would be predicted from molecular mass (not shown). Green: respect to the myofilaments that are aligned with the long axis of small, uncharged molecules; blue: ions or small charged metabolites; red: the cylindrical fibers. It is therefore not surprising that diffusion in proteins; black: dextrans; cyan: the small, fluorescent probe calcein-AM. various skeletal muscle types is also anisotropic, where radial See text for additional details. Regression line: D87.09ϫmolecular diffusion is slower than axial diffusion for small molecules like –0.69 2 mass , R 0.86, P<0.0001. Data are means ± s.e.m. D values were water (Cleveland et al., 1976; Galban et al., 2004), phosphocreatine obtained from the studies discussed in the text. (Pcr) (Moonen et al., 1990; van Gelderen et al., 1994; Kinsey et al., 1999; de Graaf et al., 2000), ATP (de Graaf et al., 2000), arginine phosphate (Kinsey and Moerland, 2002) and the fluorescent probe (Papadopoulos et al., 2000; Arrio-Dupont et al., 2000) and dextrans calcein (Hardy et al., 2009), as well as for some larger molecules (Arrio-Dupont et al., 1996) the difference between D in the like mRNAs (Dix and Eisenberg, 1988). The reduction in the radial sarcoplasm and D in water increased with molecule size, until in D is time dependent, where D declines with increasing diffusion the largest proteins D was essentially negligible. This indicates that time until 100–200ms, at which point D reaches a steady-state diffusion of large molecules in muscle is more hindered by value (Moonen et al., 1990; van Gelderen et al., 1994; Kinsey et intracellular structures such as the myofilaments than are small al., 1999; de Graaf et al., 2000; Kinsey and Moerland, 2002). In metabolites, and macromolecules or large complexes like contrast, axial D is generally independent of diffusion time. This polysomes exceeding 10–20nm in radius may be nearly immobile pattern is again characteristic of diffusion through a porous (Russell and Dix, 1992; Arrio-Dupont et al., 2000; Papadopolous medium, where as the diffusion time increases the diffusing species et al., 2000). is more likely to encounter a barrier that will impede its progress, The effect of hydrodynamic radius on globular proteins is more thus lowering D. A steady-state D is reached when the distance dramatic than in dextrans, which are thought to have a random coil traveled greatly exceeds the spatial scale of the barriers. structure that allows them to more readily access the diffusible It has been difficult to account for the barriers that induce the volume of the fiber (Arrio-Dupont et al., 2000). Nevertheless, D in pattern of diffusion observed in muscle. It has been proposed that muscle fibers is ultimately inversely related to molecular mass for restriction within the cylindrical sarcolemma is responsible for the a wide range of molecules, despite the variation in the way time-dependent anisotropy of D in muscle (Moonen et al., 1990; intracellular structures interact with molecules that differ in size van Gelderen et al., 1994). However, the same anisotropic pattern and conformation (Fig.3A) (Papadopolous et al., 2000). However, was observed in isolated fish and crustacean muscles that had very as observed by Arrio-Dupont and colleagues (Arrio-Dupont et al., large diameters, and simulations of diffusion within cylinders of 1996) for dextran diffusion, D is linearly related to the inverse of this size indicated that the sarcolemma had a negligible influence

THE JOURNAL OF EXPERIMENTAL BIOLOGY Diffusion influences in skeletal muscle 267 on the observed pattern of diffusion (Kinsey et al., 1999; Kinsey The pioneering work of August Krogh and A. V. Hill provided and Moerland, 2002). These studies also included models of equations that are still used to describe concentration profiles of O2 diffusion through the thick and thin filament lattice, and showed (Krogh, 1919) and high-energy phosphate molecules (Hill, 1965) in that the nanometer scale of these diffusion barriers caused a time- muscle. Mainwood and Rakusan (Mainwood and Rakusan, 1982) dependent decrease in D that was much too fast to account for the applied these equations to show that clustering of mitochondria near observed pattern. Based on the evidence that spatial barriers capillaries and the presence of a near-equilibrium CK reaction led intermediate in scale to the thick and thin filament array (nanometer to a smaller decrease in PO2 across the cell and less steep gradients scale) and the sarcolemma (tens or hundreds of micrometers in for PCr, ATP and ADP, and therefore helped preserve the free scale), the SR was proposed as a likely intracellular barrier that energy of ATP hydrolysis, G, across the fiber. This provided the could explain the observed pattern of anisotropy (Kinsey et al., first quantitative demonstration that the distribution of mitochondria 1999; Kinsey and Moerland, 2002). in skeletal muscle is influenced by diffusion constraints (see below). Using a different mathematical approach and assuming a A number of more elaborate mathematical models of O2 flux and cylindrical barrier to diffusion in the muscle fiber, de Graaf and metabolism have been developed for skeletal muscle (e.g. colleagues proposed that intracellular barriers with a length scale Federspiel, 1986; Groebe, 1995; Hoofd and Egginton, 1997; Piiper, of 16–22m accounted for the pattern of diffusion in muscle (de 2000; Lai et al., 2007; Dash et al., 2008), and there are a number of Graaf et al., 2000), but the identity of such a barrier is unknown reviews of aspects of O2 and aerobic substrate transport to (SR spacing is typically 1–3m). Similar results were obtained mitochondria in muscle (e.g. Hoppeler and Weibel, 1998; Wagner, from the mathematical analysis of Aliev and Tikhonov, who found 2000; Suarez, 2003; Weibel and Hoppeler, 2004). Some conclusions that a semi-permeable SR membrane shield could explain the from past work are that (1) control of O2 flux to, and usage by, experimental measurements of diffusion anisotropy, but only if the mitochondria is shared among the various steps in the O2 cascade, diameter of the shield was about 20m (Aliev and Tikhonov, (2) a substantial decrease in PO2 occurs between the capillary and 2004). More recently, Shorten and Sneyd found that they could the fiber, and (3) intracellular O2 gradients may be present. Thus, predict the observed diffusion anisotropy by modeling the influence the rate of O2 diffusion into and across the fiber to the mitochondria of thick and thin filaments, SR, t-tubules and mitochondria, may influence muscle structure and function. The intracellular O2 suggesting no role for increased cytoplasmic viscosity on D concentration gradients that are necessary for diffusion control of (Shorten and Sneyd, 2009), which is consistent with data indicating aerobic metabolism are notoriously difficult to demonstrate that the solvent viscosity of the cytoplasm is not substantially experimentally. However, spatial gradients in the redox state of different from water (see above). However, their analysis only isolated frog skeletal muscle fibers suggest that O2 gradients may characterized the steady-state pattern of diffusion and not the time influence rates of oxidative phosphorylation even under conditions dependence of D, so the intracellular structure(s) leading to the of high extracellular PO2 (Hogan et al., 2005). Further, observed anisotropy pattern in a variety of skeletal muscle types measurements of O2 consumption rates and force production in remains to be identified. isolated frog skeletal muscle fibers and rat myocardial trabeculae While the pattern of anisotropic diffusion of small metabolites suggest that maximal respiration rates cannot be attained in vivo has been observed consistently, diffusion of proteins appears to be because O2 diffusive flux is insufficient to prevent anoxia in the fiber isotropic (Maughan and Godt, 1999; Papadopoulos et al., 2000; Lin core (van der Laarse et al., 2005). et al., 2007). Shorten and Sneyd provided two potential The wide interest in intracellular O2 transport has prompted explanations for the discrepancy between small and large molecules many studies of Mb, as it is the final mediator of O2 flux to the (Shorten and Sneyd, 2009). First, barriers to axial movement, such mitochondria. Mb is a small intracellular oxygen-binding heme as the Z-line and M-line, might serve to obstruct diffusion of large protein that is found in aerobic fibers, and is thought to function molecules, while smaller molecules presumably pass through more primarily in temporal buffering of PO2 and in facilitated transport easily. Second, hydrodynamic wall effects become important for of O2 to mitochondria (reviewed in Conley et al., 2000; Jurgens et large molecules diffusing in confined spaces. Interactions of al., 2000; Wittenberg and Wittenberg, 2003; Ordway and Garry, macromolecules with intracellular structures can impede progress 2004). While its role as a temporal buffer is generally accepted, the even if the structure surface is oriented parallel to the direction of importance of the related facilitated diffusion function has been diffusion, and this hydrodynamic drag will increase as the diffusing questioned based on reaction–diffusion analyses that incorporated species approaches the size of the pores in the media. Thus, the relatively low D for Mb in muscle fibers (e.g. Jurgens et al., diffusion will become more isotropic as hydrodynamic radius 2000; Lin et al., 2007). However, Mb-facilitated diffusion becomes increases (Shorten and Sneyd, 2009). more important in skeletal muscle at low PO2 (Lin et al., 2007). Compensatory responses in the cardiovascular system that enhance Evaluating diffusion-dependent processes in muscle O2 delivery in Mb knock-out mice first reported by Gödecke and Aerobic metabolism colleagues seem to support an O2 transport role for Mb (Gödecke Measurements of D in muscle as described above, along with et al., 1999). However, Mb also has nitric oxide (NO) oxygenase microscopic analyses of diffusion distances and measurements of activity (Flögel et al., 2001), and it has been proposed that the the rates of metabolic processes, have been used to quantitatively ‘naturally occurring genetic knockout’ of Mb and Hb in Antarctic evaluate the reaction–diffusion processes in skeletal muscle shown icefishes leads to cardiovascular compensations that stem from high in Fig.1. By far, the most widely studied is aerobic metabolism, levels of NO, rather than from reduced O2 delivery per se (Sidell which depends on the diffusion of O2 to the mitochondria and the and O’Brien, 2006). Thus, while a facilitated diffusion function is subsequent diffusion of ATP to sites of utilization in the fiber an unavoidable consequence of the free diffusion of Mb and rapid (Fig.1A,B). A comprehensive review of the literature on the role reversible binding to O2, the relative importance of this transport of diffusion in aerobic metabolism is beyond the scope of this role remains the source of debate. paper, so we will briefly focus on a few examples and some of the The diffusion of ATP from the mitochondria to cellular ATPases general conclusions. has also been a subject of contention. Bessman and Geiger

THE JOURNAL OF EXPERIMENTAL BIOLOGY 268 S. T. Kinsey, B. R. Locke and R. M. Dillaman

(Bessman and Geiger, 1981) originally proposed the ‘PCr shuttle’ C (TnC), where it binds and promotes actin–myosin interactions. to explain ATP delivery from the mitochondria to cellular ATPases Relaxation of muscle requires the reuptake of Ca2+ into the SR, (Fig.1B), where the bulk of ATP-equivalent transport occurred via which is catalyzed by the SR/endoplasmic reticulum (ER) Ca2+ PCr diffusion, rather than directly as ATP. Central tenets of this ATPase (SERCA) (Fig.1C). PA is a small soluble protein that idea as it has evolved are that the mitochondrial outer membrane reversibly binds Ca2+ and is present in some muscles, with greater is a barrier for ATP/ADP diffusion, but not for PCr/Cr diffusion quantities in fast-twitch fibers. PA therefore might be expected to (although this notion has been disputed) (see Kongas et al., 2004), serve a facilitated diffusion role. However, the binding of Ca2+ by and that ATP produced in the mitochondria passes directly from PA is too slow to promote equilibrium of [Ca2+], [PA] and the adenine nucleotide translocator in the inner membrane to the [PACa2+] during a series of rapid contractions, and PA probably mitochondrial form of CK. Meyer and colleagues provided an serves as a slow-acting temporal buffer which binds the excess Ca2+ opposing view using a simple facilitated diffusion model (akin to that accumulates during consecutive contractions (Permyokov, that used to assess Mb function) to evaluate ATP-equivalent 2006). diffusive flux (Meyer et al., 1984) and concluded that (1) most Cannell and Allen first evaluated Ca2+ cycling as a ATP-equivalent diffusion should occur in the form of PCr because reaction–diffusion process in frog skeletal muscle, and these of its higher concentration and higher D, and (2) there are minimal authors generated a mathematical simulation that compared well to concentration gradients of high energy phosphates in muscle. There experimental measurements of Ca2+ transients (Cannell and Allen, has been a great deal of experimental and modeling work since 1984). A principal finding was that substantial gradients in Ca2+ these early studies that has characterized so-called phosphotransfer appear to exist over the sarcomere. Baylor and Hollingworth networks like the CK system with respect to enzyme localization, performed a similar analysis in frog skeletal muscle where they channeling of substrates, and restricted diffusion in mammalian included the influence of Ca2+ binding to ATP (Baylor and skeletal and cardiac muscle, and a number of contrasting reviews Hollingworth, 1998). In addition to finding Ca2+ gradients that are available (e.g. Walliman et al., 1992; Dzeja and Terzic, 2003; persisted for tens of milliseconds after release from the SR, they Saks et al., 2008; Beard and Kushmerick, 2009). While the function found that ATP serves to facilitate Ca2+ diffusion as ATP occurs in of the CK system is largely viewed through the lens of the high concentrations and has a relatively high D. This allowed for mammalian cardiomyocyte, Ellington provides a review of the a more uniform distribution across the sarcomere of Ca2+ that was evolution and function of the entire family of phosphagen kinases bound to TnC, which may lead to a more unified contractile that are found in animals (Ellington, 2001). response. Baylor and Hollingworth also noted that the facilitated The role of diffusion in responses of fish muscle aerobic diffusion role played by ATP makes it a temporal buffer as well metabolism to cold has been the focus of study because both (Baylor and Hollingworth, 1998). That is, ATP binding and release catalytic and diffusive processes are slower at lower temperature. of Ca2+ caused the free Ca2+ transient to be broader and of a lower Muscle from cold-acclimated or -adapted fishes typically has a high magnitude than if ATP was absent or immobilized. This model was mitochondrial volume density and high lipid composition, more recently applied to mammalian skeletal muscle, where it compared with that in warm water species. The high mitochondrial differed from the model for frog muscle in that the Ca2+ release content is thought to not only offset the reduced reaction rates at sites were offset by 0.5m from the sarcomere Z-line, based on colder temperatures but also shorten diffusion distances between morphological studies (Baylor and Hollingworth, 2007). At a mitochondria (e.g. Johnston, 1982; Egginton and Sidell, 1989) common temperature, positioning of the Ca2+ release sites near the (reviewed in Sidell, 1998). Hubley and colleagues used a middle of the thin filaments, as seen in mammals, had the advantage reaction–diffusion model of high-energy phosphates to evaluate of promoting a more uniform distribution of Ca2+ in the TnC metabolism as a function of temperature in red and white goldfish binding sites. This would seem to indicate that mammalian SR muscle, and concluded that diffusion constraints of these molecules structure has been subjected to selective pressure to offset diffusion were not the primary cause of mitochondrial proliferation in the constraints that otherwise may limit contractile function. cold (Hubley et al., 1997). However, the higher lipid content in cold Groenendaal and colleagues (Groenendaal et al., 2008) adapted water fishes, including the lipid membranes in the abundant the model of Baylor and Hollingworth (Baylor and Hollingworth, mitochondria, likely aid diffusion of O2 because of its higher 1998) to explicitly examine the differences between mammalian solubility in lipids than in the aqueous cytosol (Sidell, 1998). and frog muscle at 35°C. Again, the models implied that steep Further, the mitochondria in hemoglobin-free icefishes are larger, concentration gradients existed for Ca2+ during contractions, and have a higher lipid:protein ratio, and do not have a higher catalytic that [Ca2+] was 5-fold higher in the region of TnC than in other capacity than in warm water species, indicating that mitochondrial regions of the sarcomere in mouse and frog muscle. In addition, 2+ volume increases may serve to promote O2 diffusion in this group Groenendaal and colleagues showed that [Ca ] was high in the rather than to maintain catalytic capacity in the cold (O’Brien and region of the mitochondria, and they speculated that this Mueller, 2010). Egginton and colleagues used a reaction–diffusion arrangement facilitated Ca2+ activation of oxidative model to evaluate O2 gradients in aerobic muscle from Antarctic, phosphorylation and helped balance ATP demand during sub-Antarctic and Mediterranean species of fishes (Egginton et al., contraction (Groenendaal et al., 2008), although it should be noted 2002). Temperature had a large influence on the extent of gradients that their simulations analyzed fast-twitch fibers that rely primarily as expected, but the PO2 in the core of the fiber was inversely related on anaerobic fuels (PCr, glucose) to power rapid contractions. to fiber size across species and temperature regimes, indicating that diffusion distance is a critical parameter constraining aerobic Nuclear function design in some fish muscle. Nuclei are associated with the transcription, translation and transport of a variety of molecules ranging in size from small Ca2+ cycling metabolites to large complexes such as polysomes (Fig.1D). Muscle contraction entails the release of Ca2+ from the terminal Skeletal muscle fibers are multinucleated cells with the nuclei cisternae during muscle activation and diffusion of Ca2+ to troponin typically located at the periphery of the fiber, and each nucleus

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Fig.4. A 3-dimensional reconstruction of the microtubule array (orange) in and around nuclei (blue) from white skeletal muscle of the smooth butterfly ray (Gymnura micrura). The microtubule network may be involved in the positioning of nuclei and trafficking of nuclear products in fibers (see text). serves a volume of cytoplasm known as the myonuclear domain. myocytes microtubule-based transport is essential for the The myonuclear domain has often been considered to be conserved movement of mRNPs from the nucleus to sites of translation, and in skeletal muscle, including during increases or decreases in fiber that disruption of the microtubule system inhibits protein synthesis size (e.g. Allen et al., 1995; Roy et al., 1999; Bruusgaard et al., (Scholz et al., 2008). To our knowledge, cytoskeleton-based 2003; Bruusgaard et al., 2006; Brack et al., 2005), although this transport of proteins or mRNA has not been demonstrated in mature notion does not seem to be generally applicable and remains the skeletal muscle fibers. However, the microtubule network is closely source of debate (reviewed in Gundersen and Bruusgaard, 2008). associated with nuclei in mature skeletal muscle from mammals Nevertheless, the size of the myonuclear domain presumably is (e.g. Bruusgaard et al., 2006), crustaceans and fishes (Fig.4), and regulated to ensure sufficient transcriptional capacity as well as it appears to be well suited for both positioning of nuclei (perhaps limited distances over which nuclear substrates and products must playing a role in controlling myonuclear domain size) and travel to reach sites of action. There is evidence that transport transporting nuclear products. within the myonuclear domain governs nuclear distribution at the sarcolemma, as mathematical analyses indicate that myonuclei in Comparative analyses of muscle growth reveal mechanisms mammalian skeletal muscle have a uniform distribution at the of avoiding diffusion limitation sarcolemma (rather than a random distribution), which minimizes While the vast majority of work on reaction–diffusion processes in transport distances within the domain. This suggests that there muscle has focused on mammals, our labs have for the past several are through-space signals that influence nuclear positioning years examined muscles from non-traditional models that have (Bruusgaard et al., 2003; Bruusgaard et al., 2006). properties that make them particularly informative. As described The capacity for movement of nuclear products appears to be above, the extent to which diffusion influences metabolic processes highly constrained in skeletal muscle, and proteins tend to remain is largely dependent on the interaction between reaction fluxes and in the vicinity of the nucleus from which they originated (Hall diffusion distances (Fig.2), and during animal growth this and Ralston, 1989; Pavlath et al., 1989; Ono et al., 1994). The interaction can be altered. As an animal increases in body mass, distribution of mRNA and messenger ribonucleoprotein particles muscles grow by a combination of increasing the number of fibers (mRNPs) in skeletal muscle is also consistent with greatly hindered (hyperplasia) and increasing the size of individual fibers diffusion, and it has been suggested that these large complexes are (hypertrophy). Thus, hypertrophic fiber growth leads to increasing excluded from the myofibrillar space (Russell and Dix, 1992). diffusion distances for molecules like O2. Further, in animals However, Gauthier and Mason-Savas found ribosomes (and such as fishes and crustaceans that undergo a very large post- possibly polysomes) within the thick and thin filament lattice metamorphic increase in body mass and have muscles that grow (Gauthier and Mason-Savas, 1993), suggesting that these large hypertrophically, fibers can greatly increase in diameter during complexes have access to this region and may promote localized animal growth. For instance, some juvenile crustaceans and fishes translation. It is also possible that diffusion constraints may be have muscle fibers that are <50m in diameter, but as the animals overcome in part by the extensive microtubule array in skeletal grow the fibers can increase in diameter to several hundred muscle, which is closely associated with the nuclei (Bruusgaard et micrometers (Jahromi and Atwood, 1971; Hoyle, 1987; Weatherley al., 2006) and may be used in the trafficking of mRNA and proteins. and Gill, 1987; Boyle et al., 2003; Johnston et al., 2003; Johnston A dynamic, anti-parallel microtubule network develops in growing et al., 2004; Kinsey et al., 2007; Nyack et al., 2007; Jimenez et al., myotubes, and it has been shown that myosin can be transported 2008; Jimenez et al., 2010; Hardy et al., 2009; Hardy et al., 2010). along these filaments by motor proteins (Pizon et al., 2005). Scholz Thus, for processes where fiber size constitutes an important and colleagues demonstrated that in fully differentiated cardiac diffusion distance (as it does for aerobic metabolism and nuclear

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A as data from the literature were evaluated using a reaction–diffusion Many fibers lie near model and the application of the effectiveness factor (), which has diffusion limitation in adults typically been used by engineers to assess the design of catalytic reactors (Weisz, 1973).  is the ratio of the observed reaction rate to the rate if diffusion were not limiting, and this ratio can be calculated if the diffusion coefficients, diffusion distances and reaction rates are known. Therefore, if 1, diffusion is not limiting, whereas if it is 0.5, the observed rate is 50% of what it would be if diffusion was not constraining the reaction. Our models are admittedly simple representations of fibers, treating them as bioreactors and not accounting for all kinetic regulatory mechanisms, enzyme localization or potential channeling of Aerobic ATP substrates. However, they allow us to ask the general question in a rate (mmol l wide range of muscle types: for a given set of D values, diffusion turnover adius (µm) distances and rates of ATP turnover, does diffusion limit aerobic –1 min Fiber r –1 flux? Using this approach, we have found that aerobic flux in ) muscle fibers typically is not diffusion limited, but fibers are often on the brink of substantial limitation (Kinsey et al., 2007; Locke B and Kinsey, 2008; Jimenez et al., 2008; Hardy et al., 2009; Dasika et al., 2011). That is, many muscles have a combination of ATP turnover rates and diffusion distances that maintain a high , but if aerobic flux or diffusion distances were to increase further, a substantial drop in  would occur. Further, in cases where we have examined muscles that grow hypertrophically, we found that fibers only approach diffusion limitation in adults, suggesting that diffusion may play a role in muscle structure and growth patterns (see below; Fig.5). It is interesting to note that Weisz originally suggested that biological reaction–diffusion processes would function near the edge of diffusion limitation (Weisz, 1973) and we have found that this often occurs in skeletal muscle. Aerobic ATP Skeletal muscle also is subject to metabolic and structural rate (mmol l turnover adius (µm) alterations to avoid diffusion limitation. A particularly informative –1 min Fiber r –1 model has been the swimming muscle of the blue crab, Callinectes ) sapidus, which has fibers that increase in size during hypertrophic growth from <50m in juveniles to >600m in adults (Boyle et Fig.5. Example of the interaction between the effectiveness factor (), al., 2003). These muscles are composed of light fibers, which are aerobic ATP turnover rate and fiber radius for (A) a high blood [O2] –1 –1 used for burst contractions and have sparse mitochondria, dark (35moll ) and (B) a low blood [O2] (8moll ). When <1, diffusion of O2 or intracellular metabolites like ATP limits aerobic reaction flux. This fibers, which are rich in mitochondria and power sustained surface was modeled for fibers that have 10% of the cell volume devoted swimming, and a small number of intermediate fibers. As diffusion to mitochondria. In principle, fibers with a similar mitochondrial volume distances increase with animal growth, the light and dark fibers density can be designated as points on the surface, and changes in fibers respond in different ways to avoid diffusion limitation. As the light (e.g. hypertrophic growth) can be represented by contours on the surface. fibers grow hypertrophically, they rely increasingly on anaerobic All fibers examined so far have aerobic fluxes and diffusion distances that metabolism to speed up key phases of post-contractile recovery. place them on the upper part of the surface under aerobic conditions That is, the small fibers from juveniles produce no lactate and (Ϸ1), but many are near the edge of diffusion limitation, particularly as consume little glycogen after burst contraction, but the large fibers adults (not shown). Note that a reduced blood [O2] makes fibers more susceptible to diffusion limitation. of adults produce copious amounts of lactate and consume glycogen post-contraction (Boyle et al., 2003; Johnson et al., 2004). This allows phosphagen resynthesis to occur faster than our reaction–diffusion model predicted that it should in the large fibers function), muscles from these animals demonstrate natural (Kinsey et al., 2005). Of course, this puts the animal deeper into variation in one of the key properties that governs diffusion- oxygen debt, but it has the immediate advantage of allowing a more dependent reaction fluxes. rapid recovery of high-force contractile function. Previously measured values of diffusion in muscle (see above) The light fibers also undergo structural modifications to were coupled with measurements of diffusion distances determined compensate for hypertrophic growth. As fibers get larger, the using confocal and transmission electron microscopy, and mitochondria shift from a uniform distribution with both peripheral measurements of aerobic ATP turnover. Rates of aerobic ATP (subsarcolemmal) and central (intermyofibrillar) mitochondria in turnover were determined from measurements of mitochondrial juveniles, to an almost exclusively subsarcolemmal distribution in volume density (Kinsey et al., 2007; Locke and Kinsey, 2008), adults, where nearly 90% of the mitochondria are clustered at the from respiration rates of isolated mitochondria (Burpee et al., sarcolemma (Boyle et al., 2003; Hardy et al., 2009). Thus, the 2010), or from direct measurements of ATP turnover using NMR mitochondria are moving closer to the source of the substrate, O2, in isolated muscles or in vivo (Kinsey et al., 2005; Hardy et al., but this creates diffusion distances of hundreds of micrometers for 2006; Nyack et al., 2007; Jimenez et al., 2008). These data as well the product, ATP, in these very large fibers (Fig.1A,B). This

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Blue crab light fibers: extreme hypertrophy Fig.6. Patterns of post-metamorphic fiber growth and and organelle shifts organelle distribution that lead to an avoidance of diffusion limitation during animal growth. Green: mitochondria; blue: A nuclei. (A)Blue crab light muscle. As the fibers grow hypertrophically through most of post-metamorphic development to very large sizes, mitochondria shift from a uniform distribution (subsarcolemmal and intermyofibrillar) toward the fiber periphery (subsarcolemmal) and nuclei do the opposite. These fibers also rely increasingly on anaerobic metabolism to increase post-contractile recovery during fiber growth. (B)Blue crab dark muscle. The fibers Blue crab dark fibers: extreme hypertrophy become increasingly subdivided during hypertrophic and fiber subdivision growth, and develop intracellular perfusion. The subdivisions are isolated from one another and represent B the metabolic functional unit, while the fiber as a whole represents the contractile functional unit. (C)Black sea bass white muscle. The muscles grow hypertrophically through most of post-metamorphic development and undergo a shift in mitochondrial and nuclear distribution as in blue crab light muscle. However, as the fibers approach Black sea bass white fibers: hypertrophy and diffusion limitation of aerobic metabolism, there is an onset organelle shifts, followed by hyperplasia/fiber splitting of hyperplasic growth that leads to shorter diffusion distances. (D)Typical pattern of fish white muscle growth. C Hyperplasia is a dominant form of muscle growth early in life (often accompanied by hypertrophy), followed by nearly exclusive hypertrophic growth. The duration of the hyperplasic growth period will impact the fiber size in adults. This pattern is similar in fish red muscle and in mammalian muscle, but the fibers do not grow as large as Typical fish white fibers: mixture of hyperplasia and in white muscle. (E)White muscle from sharks and rays. hypertrophy followed by hypertrophy and organelle Mitochondria shift distribution during hypertrophic growth shifts as in other muscle, but the nuclei have a largely intermyofibrillar distribution, even when the fibers are D small. Here, the nuclei are not responding to changing diffusion constraints, but are prepared for impending constraints in the adult. (F)Red muscle from sharks and rays. The red muscles undergo a modest shift in mitochondria, but the nuclei are subsarcolemmal throughout fiber growth. Unlike the white fibers in E, the Shark and ray white fibers: hypertrophy with shifts red fibers will not grow large (due to diffusion constraints in mitochondria and persistent intermyofibrillar nuclei associated with aerobic metabolism) and a subsarcolemmal nuclear placement is therefore not E problematic. See text for additional details.

Shark and ray red fibers: limited hypertrophy with modest mitochondrial shifts and persistent F subsarcolemmal nuclei

suggests that O2 diffusion is a greater constraint than is ATP However, as the fibers grow to large sizes, not only are more nuclei diffusion. Reaction–diffusion models demonstrated that this shift recruited to the fiber but also they become uniformly distributed in distribution allows the muscle to maintain a high  and achieve throughout the fiber (subsarcolemmal and intermyofibrillar). This the experimentally measured rate of aerobic ATP turnover, whereas also appears to be a response to diffusion constraints. Nuclei do not if the mitochondria had maintained a uniform mitochondrial require O2 like mitochondria, but they rely on the transport of distribution throughout fiber growth, the fibers would be highly macromolecules that diffuse very slowly in muscle (see above). By diffusion limited (very low ) in adults and they would attain only both increasing the nuclear number and altering the distribution, the a fraction of the observed rate of aerobic flux (Hardy et al., 2009). fibers preserve both the size of the myonuclear domain and the In contrast to mitochondria, nuclei undergo a shift in distribution maximal diffusion distances within that domain. Reaction– during hypertrophic growth that is the opposite of that seen for diffusion analyses again reveal that rates of nuclear processes are mitochondria. In small fibers of juveniles, nuclei have a three orders of magnitude greater than they would be if this subsarcolemmal distribution, as is typical of mammalian fibers. distributional change had not occurred (Hardy et al., 2009). The

THE JOURNAL OF EXPERIMENTAL BIOLOGY 272 S. T. Kinsey, B. R. Locke and R. M. Dillaman structural changes seen in blue crab light fibers are shown Conclusions schematically in Fig.6A. Skeletal muscle fibers are highly ordered and contain many The blue crab also has aerobic (dark) fibers that power intracellular barriers that impede diffusion, and measurements of sustained swimming, which require much higher rates of aerobic diffusion in muscle for a wide range of molecules indicate that the ATP turnover and respond very differently from the burst (light) intracellular environment behaves as a porous medium. This does fibers. These fibers become increasingly subdivided as the fibers not discount the importance of enzyme localization, specific grow larger, creating both short intracellular diffusion distances binding or transport along the cytoskeleton, all of which may be during growth and intrafiber perfusion. Within a subdivision, mechanisms to avoid diffusion limitation of certain processes. mitochondrial and nuclear distributions are very reminiscent of a Examination of a variety of species and muscle types with fibers mammalian fiber, and FRAP experiments demonstrated that that undergo hypertrophic growth, and application of simple neighboring subdivisions do not exchange small molecules. Thus, reaction–diffusion mathematical models indicate that most fibers the subdivisions represent independent metabolic functional units are not limited by diffusion per se. However, many fibers seem to of the fiber. However, innervation patterns revealed that the fiber be near diffusion limitation, particularly in adults, and changes in as a whole remains the contractile functional unit. Thus, the metabolic organization and growth patterns appear to be aimed at aerobic fibers have solved design constraints by reducing avoiding diffusion constraints. This implies that intracellular diffusion distances for aerobic metabolism, which is constrained signals that govern processes like mitochondrial biogenesis/ by diffusion, but preserving the structures associated with degradation or nuclear position are spatially variable across the cell, contraction, which is not limited by diffusion (Fig.6B). and changes in fiber growth can be altered by signals associated Mathematical reaction–diffusion models again showed that all of with diffusion limitation. the changes described above in the aerobic fibers are essential for the muscle to support the observed rates of ATP turnover (Hardy Acknowledgements et al., 2009). This research was supported by grants from the National Science Foundation (IOS-0719123 to S.T.K. and R.M.D. and IOS-0718499 to B.R.L.) and from the We have since shown that these design features in light and National Institute of Arthritis and Musculoskeletal and Skin Diseases (R15-AR- dark muscle from a variety of swimming and non-swimming 052708 to S.T.K.). Deposited in PMC for release after 12 months. crabs are generally applicable and independent of phylogeny. This indicates that they are adaptive responses to diffusion References constraints that depend on fiber (and body) size and functional Agutter, P. S., Malone, P. C. and Wheatley, D. N. (1995). Intracellular transport demand (Hardy et al., 2010). We have seen the same shifts in mechanisms: a critique of diffusion theory. J. Theor. Biol. 176, 261-272. Aliev, M. K. and Tikhonov, A. N. (2004). 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